Part 1:
Refer to attached file (IMG_0201.JPG) for diagram:
Area of centre triangle = 1/2 base x height = 1/2 b x a sin(C) = 1/2 ab sin(C).
or 1/2 c x a sin(B) = 1/2 ac sin(B)
or 1/2 b x c sin(A) = 1/2 bc sin(A)
Area of triangle between squares a and b = 1/2 b x a sin(C) = 1/2 ab sin(C).
Area of triangle between squares a and c = 1/2 c x a sin(B) = 1/2 ac sin(B).
Area of triangle between squares b and c = 1/2 b x c sin(A) = 1/2 bc sin(A).
Part 2:
Draw a line between A and C -
Triangle ABC = triangle BGH - as proven in part 1
Triangle ACD = triangle DKL
ABC + ACD = BGH + DKL
Draw a line between B and D -
Triangle ABD = triangle AEF
Triangle BCD = triangle CIJ
ABD + BCD = AEF + CIJ
ABC + ACD = ABD + BCD
Therefore, BGH + DKL = AEF + CIJ
Part 3:
Triangle ABC = triangle CIL - as proven in part 1
Draw a line between C and D -
Triangle ACD = triangle AFK
Triangle BCD = triangle BGJ
Therefore, ABC = AFK + BGJ
Therefore, CIL = AFK + BGJ